The following are the guidelines, not rules, that I gathered from this (somewhat controversial) book titled 日本語に主語はいらない—百年の誤謬を正す.
/ari/ as an intransitivity marker
If one of the pair ends with /ari/ in its ます-stem, it's intransitive. This intransitivity marker /ari/ is historically related to ある (to be). The unmarked (i.e. transitive) side of the pair mostly (if not always) ends with /e/ in its ます-stem.
I list all examples below in their ます-forms so the marking is clear, and also put an asterisk on the marked side. Each line has the intransitive verb first followed by the transitive counterpart.
- 上がります* vs. 上げます
- 下がります* vs. 下げます
- 閉まります* vs. 閉めます (締まります* vs. 締めます)
- 始まります* vs. 始めます
- 変わります* vs. 変えます
- 集まります* vs. 集めます
- 教わります* vs. 教えます
- 預かります* vs. 預けます
- 受かります* vs. 受けます
- 見つかります* vs. 見つけます
There are exceptions to this rule. 割る is one such case. It is probably etymologically unrelated to ある, and it's the unmarked side in its pair (with 割れる). I will come back to these pairs later.
I don't know if 積もる, with /ri/ but not /a/, is a variation of this pattern or unrelated.
/si/ as a transitivity marker
If one of the pair ends with /si/ (し) in its ます-stem, it's transitive. This transitivity marker /si/ is historically related to する (to do). The unmarked (i.e. intransitive) side of the pair may or may not end with /e/ in its ます-stem.
- 落ちます vs. 落とします*
- 起きます vs. 起こします*
- 通ります vs. 通します*
- 戻ります vs. 戻します*
- 出ます vs. 出します*
- 消えます vs. 消します*
- 隠れます vs. 隠します*
- 汚れます vs. 汚します*
- 壊れます vs. 壊します*
In some of the pairs in this category, the transitive side is actually the short (old) causative form. 飛ばす is one of them.
- 沸きます vs. 沸かします*
- 動きます vs. 動かします*
- 飛びます vs. 飛ばします*
The /s/ sound in causative forms is also thought to be historically related to する.
If neither of the pair is marked by /ari/ or /si/, one ends with /i/ and the other with /e/ in their ます-stems. Between them, the one with /e/ is considered marked and the one with /i/ unmarked. The verbs in this category are further divided into two subgroups depending on which direction the /e/ marker works in.
A. /e/ as a transitivity marker
- 開きます vs. 開けます*
- 立ちます vs. 立てます*
- 付きます vs. 付けます* (also written with other kanji)
- 続きます vs. 続けます*
- 並びます vs. 並べます*
- 届きます vs. 届けます*
- 育ちます vs. 育てます*
- 縮みます vs. 縮めます*
B. /e/ as an intransitivity marker
- 割れます* vs. 割ります
- 焼けます* vs. 焼きます
- 切れます* vs. 切ります
- 破れます* vs. 破ります
- 折れます* vs. 折ります
- 砕けます* vs. 砕きます
To determine which group a verb belongs to, we need to rely on semantics. The verbs in the second group are break-like verbs. The unmarked (default) side, of which 割る is one, denotes a forceful and destructive action upon something, and therefore transitive, and the marked side (i.e. with /e/) describes a change that results from it, and therefore intransitive.
The book argues transitivity in Japanese is not a dichotomy but a continuum of how much control the speaker has over the action or event of the verb. At the least transitive end are passive forms. The /(r)are/ morpheme in passive forms is historically related to ある. At the other end are causative forms with the /(s)ase/ morpheme, whose /s/ sound is historically related to する. (You have control over what other people do.) The /ari/ and /si/ markers above, which are thought to be the results of (the old forms of) ある and する fusing into the base verbs, put the verbs inside of, respectively, the passive and causative extremes on the transitivity scale.
見える doesn't make up a pair with 見る, and neither does 聞こえる with 聞く. However, they are similar in formation to the verbs in the first subgroup of the third category above. The /e/ marker (from ゆ) here denotes spontaneity and thus puts the verbs on the intransitive side of their respective base versions.
Though not exactly intransitive-transitive pairs, godan verbs (or u-verbs) and their potential forms may be seen in a similar light. The /e/ marker, denoting potentiality, puts the verbs with it on the intransitive side of their unmarked counterparts. This may explain the use of が with potential verbs.
見せる, with /s/, clearly has a causative sense. It doesn't make up a pair with 見る, either, but must be related to the second category. (In Kansai-ben, we say 見して as well as 見せて.)